Thread: Sum and products of series

Hello,

I was wondering if anyone could help me show the following:

$\displaystyle \displaystyle\sum_{i=1}^n\frac{(t+n-1)!(i-1)!}{(t+i-1)!}=\frac{(n+t-1)!-n!}{(t-1)!(t-1)}$,

where $\displaystyle t\geqslant2$.

It's an important part of a larger problem I'm working on and I'm a little stuck.

Thanks in advance.

Originally Posted by hairymclairy

Hello,

I was wondering if anyone could help me show the following:

$\displaystyle \displaystyle\sum_{i=1}^n\frac{(t+n-1)!(i-1)!}{(t+i-1)!}=\frac{(n+t-1)!-n!}{(t-1)!(t-1)}$,

where $\displaystyle t\geqslant2$.

It's an important part of a larger problem I'm working on and I'm a little stuck.

Thanks in advance.

Induction on n.

If we checking this formula when n=1:

{(t+1-1)!(1-1)!}/{(t+1-1)!}=t!/t!=1

And:

{(1+t-1)!-1!}/{(t-1)!(t-1)}={t!-1}/{t!-1}=1

Try to continue...

I know it's provable by induction, however I need to derive the right hand side from the left, not just prove it if you see what I mean. Maybe the problem should be rephrased to "find a closed expression (i.e. an expression not involving the summation and product symbols) for the left hand side of that equation".

The question asked me to derive various formulas, but when I spoke to my supervisor he said that it's apparently fine to just find a formula and then prove by induction, so thanks!